 Hi and welcome to the session. Let us discuss the following question. Question says, in an AP given nth term that is an is equal to 4, common difference that is d is equal to 2, sum of n terms that is Sn is equal to minus 14, find n and a. First of all let us understand that sum of n terms is given by n upon 2 multiplied by 2a plus n minus 1 multiplied by d, where a is the first term of AP, Sn is the sum of n terms of AP, d is the common difference. This is the key idea to solve the given question. Let us now start the solution. We are given nth term that is an is equal to 4, common difference of AP is equal to 2 and sum of n terms of AP is equal to minus 14. All these three terms are given in the question. Now we know nth term of AP that is an is equal to a plus n minus 1 multiplied by d, where a is the first term of AP and d is the common difference. Now we know an is equal to 4, so we will substitute 4 for an. This is equal to a plus n minus 1 multiplied by 2, we know common difference is equal to 2, so we have substituted 2 for d. Now simplifying we get 4 is equal to a plus 2n minus 2, multiplying these two brackets we get 2n minus 2. Now adding 2 on both the sides we get 6 is equal to a plus 2n, subtracting 2n from both the sides we get 6 minus 2n is equal to a or we can write a is equal to 6 minus 2n. Now let us name this expression as 1. Now we know sum of n terms of AP is equal to n upon 2 multiplied by 2a plus n minus 1 multiplied by d. Now we know Sn given to us is equal to minus 14, so we can substitute minus 14 for Sn. This is equal to n upon 2 multiplied by 2a plus n minus 1 multiplied by 2, we know common difference is equal to 2. Now this implies minus 14 is equal to n upon 2 multiplied by 2a plus 2n minus 2, multiplying these two brackets we get 2n minus 2. Now this implies minus 14 is equal to n plus n square minus n, multiplying n upon 2 by this bracket we get this expression. Now let us name this expression as 2. Now substituting this value of a from expression 1 in expression 2 we get minus 14 is equal to 6 minus 2n multiplied by n plus n square minus n. Now this implies minus 14 is equal to 6n minus 2n square multiplying these two brackets we get these two terms plus n square minus n. Now we get minus 14 is equal to you know 6n minus n is equal to 5n and minus 2n square plus n square is equal to minus n square. Now adding n square minus 5n on both the sides we get n square minus 5n minus 14 is equal to 0. Now minus 5n can be written as minus 7n plus 2n so we can write n square minus 7n plus 2n minus 14 is equal to 0. Now taking n common from first two terms we get n multiplied by n minus 7 taking two common from last two terms we get 2 multiplied by n minus 7 is equal to 0. Now n minus 7 is common in these two terms so taking n minus 7 common from these two terms we get n minus 7 multiplied by n plus 2 is equal to 0. Now simplifying we get n minus 7 is equal to 0 or n plus 2 is equal to 0. Here if we add 7 on both the sides we get n is equal to 7. Here if we subtract 2 from both the sides we get n is equal to minus 2. We know number of terms can never be negative so we will neglect n is equal to minus 2 neglecting n is equal to minus 2 we get n is equal to 7. Now substituting value of n is equal to 7 in expression 1 we get a is equal to 6 minus 2 multiplied by 7. Now simplifying we get a is equal to minus 8 so our required answer is n is equal to 7 and a is equal to minus 8. This completes the session hope you understood the session take care and have a nice day.